Download Fiscal Multiplier in a Liquidity Constrained New Keynesian Economy∗

Fiscal Multiplier in a Liquidity Constrained
New Keynesian Economy
Engin Karayand Jasmin Sinz
February 12, 2014
Abstract
We study the e¤ects of …scal policy on the macroeconomy using a
liquidity constrained New Keynesian model in which private …nancial
assets are regarded as partially liquid and government bonds as liquid.
We …nd that the government spending multipliers in this economic
environment are large enough for …scal policy to be highly e¤ective.
Fiscal expansion stimulates output growth as higher public spending
creates real aggregate demand and an increase in public borrowing improves liquidity by increasing the proportion of liquid assets in private
sector wealth.
Keywords: DSGE Models, Monetary Policy, Fiscal Policy, Liquidity Trap, Credit Constraints
JEL: E32, E52, E58, E62
We are grateful to Edmund Cannon, Jonathan Temple, James Cloyne and Michael
McMahon for their helpful and constructive comments. We also thank seminar participants
at the Bank of England, University of Bath, University of Bristol, the 45th Annual Money
Macro and Finance Conference and the Royal Economic Society Easter School 2013 for
their helpful comments.
y
University of Bristol. Email: [email protected].
z
University of Bristol. Email: [email protected].
1
1
Introduction
Over the last decade, in many if not all developed countries including the
Euro-Area, the US and the UK, monetary policy has been the main instrument for managing the level and rate of growth of aggregate demand and
in‡ationary pressure. The chief monetary policy tool has been short-term
interest rates. The response to the recent …nancial crisis has typically been
lowering the short-term nominal interest rate to its zero lower bound (i.e.
generating a liquidity trap). Monetary policy loses its power at the zero
lower bound, so the conventional policy option of reducing interest rates is
no longer available. The ine¤ectiveness of monetary policy at the zero lower
bound raises the question of whether …scal policy can be used to mitigate
the e¤ects of the …nancial crisis.
Answering this question requires a model that can capture, at the very
least, the key aspects of the crisis. As many noted, the realisation at the
onset of the crisis that many private …nancial assets were of a lower quality
and therefore accompanied by higher default risks than previously assumed
led to a ‡ight to liquid assets. The asset markets experienced a severe shortage of liquid assets. At the height of the crisis, the …nancial markets for
private assets essentially froze. The drop in the resaleability of private assets
diminished …rms’ability to raise funds and use their assets as collateral in
borrowing. The consequent decrease in the funds available for investment
led to substantial drops in output and in‡ation. To combat the recession,
central banks lowered the nominal interest rate to its zero lower bound.
The aim of this paper is to study the e¤ectiveness of …scal policy using a model that is capable of capturing the scenario described above. To
this end, we use the model proposed by Del Negro, Eggertsson, Ferrero and
Kiyotaki (2011) (henceforth “DEFK”). This model reformulates the state-ofthe-art version of New Keynesian economics, as in Christiano, Eichenbaum
and Evans (2005) (“CEE”) and Smets and Wouters (2007) (“SW”), by incorporating the liquidity frictions as described in Kiyotaki and Moore (2008).
2
In the DEFK model, the economy is populated with a large number of
identical households. Each household consists of two types of members (entrepreneurs and workers) and can save in two types of …nancial assets (government bonds and private equity). The two types of …nancial assets di¤er
in the degree of liquidity. Government bonds are liquid, while private assets are not.1 During each period, a randomly chosen fraction of household
members becomes entrepreneurs. Entrepreneurs have the opportunity to invest in new capital, which gives a better return than holding government
bonds or private equity. Although investment opportunities are attractive,
entrepreneurs are liquidity constrained. They are bound by both borrowing
constraints and resaleability constraints on their equity holdings: Entrepreneurs can borrow by issuing new equity but the amount that they can issue
in any given period is limited. Private equity is illiquid, so entrepreneurs
can sell only up to a certain proportion of their equity holdings in any given
period. Workers, on the other hand, do not have the opportunity to invest
in capital and are not liquidity constrained. They work, consume and save
by holding government bonds and private equity. Holders of bonds or equity
earn a premium on top of the standard returns because these assets relax
the liquidity constraints when investment opportunity arrives in the future.
Other features of the model are standard New Keynesian. Firms and workers
enjoy some degree of monopoly power; prices and wages remain unchanged,
on average, for several months. The central bank sets monetary policy using
a Taylor-style rule.
The assumption of the model that government bonds are more liquid than
private assets is consistent with the empirical evidence provided by Krishnamurthy and Vissing-Jorgensen (2012). These authors suggest that Treasury
bonds are money-like in the sense that they can be used as a collateral or
medium of exchange in …nancial markets. They …nd that the yield spread
1
As noted by DEFK, private equity has a broad de…nition in this model. It can be
interpreted as privately issued paper such as commercial paper, bank loans, mortgages,
and so on.
3
between Treasury bonds and less liquid assets reduces when the supply of
Treasury bonds is abundant. In the theoretical literature, there are also
papers that treat government bonds as liquidity (see, for example, Woodford (1990) and Aiyagari and McGrattan (1998)). Furthermore, the presence
of the resaleability constraints in the DEFK model allows us to capture a
phenomenon in the recent …nancial crisis that many private …nancial assets
become almost illiquid. Comparison of the empirical data and the model’s
simulations shows that the DEFK model performs well in explaining the responses of the key macroeconomic variables to the recent credit crisis. We
therefore believe that the model captures the essence of the crisis.2
We introduce a role for government spending in the DEFK model and
use it to determine the value of the …scal multiplier. Since we examine
crisis situations in which the economy may spend a long time away from
the deterministic steady state, we numerically simulate the original nonlinear model without log-linearising it. Consequently, the accuracy of our
results does not depend on the equilibrium’s vicinity to the steady state. In
our experiments, we consider a bond-…nanced …scal expansion, in which the
government issues bonds to households to be repaid by tax increases at a
later date.
We carry out our analysis in two steps. In the …rst part, we examine the
value of the …scal multiplier using the version of the DEFK model at normal
times (i.e., without liquidity shocks) when the zero lower bound on the nominal interest rate does not bind. We …nd that the value of the …scal multiplier
is much greater than that suggested by a standard DSGE model without …nancial frictions. The cumulative multiplier obtained using the DEFK model
is 1.6, while it is 0.55 in the standard model. The intuition for this result
is as follows. In both models, an increase in government spending leads to
higher future tax burdens and rises in the real interest rate. Both of these
2
DEFK use their model to examine the e¤ectiveness of quantitative easing and …nd
this to be an e¤ective policy. Ajello (2010), Dri¢ ll and Miller (2011) and Shi (2012) also
use the KM framework to study the current …nancial crisis.
4
factors cause households to postpone their consumption and increase their
savings by holding more liquid government bonds. In the standard model,
investment falls since the higher real interest rate increases the opportunity
cost of investing in physical capital, thus the …scal multiplier is less than 1. In
the DEFK model, the multiplier is large for two reasons. First, unlike in the
standard model, a bond-…nanced government spending expansion improves
liquidity by increasing the proportion of liquid assets in households’wealth,
which in turn allows credit constrained entrepreneurs to increase their investment. Increased economic activity then increases private consumption.
Second, an increase in government spending creates real aggregate demand
which drives up output. We show that both of these channels are important
in generating a large …scal multiplier.
In the second part, we look at the …scal multiplier in a credit crisis
caused by a tightening of the resaleability constraints, in which case the
zero lower bound on the nominal interest rate becomes binding.3 We …nd
that, in both the DEFK and the standard models, the multiplier is much
larger in a liquidity trap than at normal times. Moreover, in the liquiditytrap case, the multiplier in the DEFK model is still larger than that in the
standard model. Under this scenario, the cumulative multiplier suggested by
the DEFK model is around 2. The …scal multiplier is larger in a liquidity trap
because an increase in government spending creates in‡ationary pressures
which decrease the real interest rate and stimulate consumption and output.
In the DEFK model, the …scal multiplier is even larger in a liquidity trap
because the multiplier e¤ect applies to both consumption and investment.
Finally, holding the persistence of government spending constant, we show
3
Erceg and Linde (2012) criticise the assumption of an exogenous zero-bound condition
in the study of the …scal multiplier. They point out that as an increase in government
expenditure may help push the economy out of a liquidity trap, the multiplier will be
smaller if the zero-bound condition is endogenous. Mertens and Ravn (2010) warn that
the value of the multiplier is sensitive to the type of shock that drives the economy into a
liquidity trap. To address these issues, we examine the …scal multipliers using the DEFK
model, in which the liquidity trap is endogenously caused by a …nancial crisis.
5
that the …scal multiplier in the standard model tends to decrease as the crisis
prolongs, whereas in the DEFK model it increases.
Before describing the details of the model, let us brie‡y review the literature on this topic.4 Most of the recent theoretical discussions on the effectiveness of …scal policy have been based on the CEE/SW model (see, for
example, Christiano, Eichenbaum and Rebelo (2011), Cogan et al. (2010)
and Woodford (2011)). The CEE/SW model assumes frictionless …nancial
markets and therefore cannot provide a detailed account of the crisis. Cogan
et al. (2010) …nd that the …scal multiplier is less than one and that this is
true even when the economy stays at the zero lower bound for a prolonged
period of time. Christiano et al. (2011) and Woodford (2011) show that the
conclusion reached by Cogan et al. (2010) - in brief, …scal policy is ine¤ective
in a liquidity trap - changes if the …scal expansion lasts exactly as long as the
zero-bound state. They …nd a …scal multiplier considerably in excess of one.
As we show later in this paper, the …scal multiplier in the DEFK model is
much larger than that suggested by a standard model similar to the one employed in Christiano et al. (2011), both in times of crisis and at normal times.
The main reason for this result is that in the standard model, the increase in
output in response to a government spending shock is almost entirely driven
by consumption and the multiplier e¤ect on investment is negligible, if not
negative. In the DEFK model, by contrast, investment is crowded in by
government spending due to the increase in the supply of liquid government
bonds. Our result that public debt can serve as private liquidity is in line
with Woodford (1990), who obtains similar …ndings using a static model.
Our paper belongs to the recent literature that examines the e¤ects of
…scal policy in the presence of …nancial frictions. Important papers in this lit4
The majority of empirical research in this area seems to suggest that …scal policy is not
e¤ective and that an increase in government spending does not have a signi…cant e¤ect
on the economy (see, for example, Hall (2009), Ramey (2011) and references therein).
The government spending multiplier is typically estimated to lie between 0.6 and 1.2.
However, some recent empirical studies show that the …scal multiplier is much larger
during a recession (see, for example, Auerbach and Gorodnichenko (2012)).
6
erature include Carrillo and Poilly (2013), Eggertsson and Krugman (2012)
and Fernandez-Villaverde (2010). Eggertsson and Krugman (2012) use a
model in which some agents’ability to borrow diminishes following a shock
similar to a “Minsky moment”. They …nd that, under such a shock, the government spending multiplier can be much greater than one, though its exact
value depends heavily on the share of debt-constrained borrowers in the economy. Carrillo and Poilly (2013) and Fernandez-Villaverde (2010), on the other
hand, use models that accommodate the type of credit frictions suggested by
Bernanke, Gertler and Gilchrist (1999) (“BGG”), in which …rms’ability to
borrow is determined by the market value of their net worth. FernandezVillaverde (2010) considers the government spending multiplier at normal
times, whereas Carrillo and Poilly (2013) focus on the case of a liquidity
trap. Fernandez-Villaverde (2010) …nds that the value of the impact multiplier is around one following the government spending shock and decreases
quickly thereafter. His multiplier is larger than that suggested by standard
models but smaller than ours.5 Carrillo and Poilly (2013) …nd that …nancial frictions have a greater contribution to the value of the multiplier in a
liquidity trap than at normal times. Indeed, their cumulative multiplier in
the liquidity-trap case is 3.7,6 which is almost twice as large as ours. Our
paper di¤ers from previous studies in the way that …nancial frictions are
introduced. While the BGG model focuses on borrowing constraints, the
DEFK model incorporates the Kiyotaki and Moore (2008) framework, which
accounts for both borrowing constraints and resaleability constraints.7 It is
5
As shown later in our results, although our post-shock impact multiplier at normal
times is smaller than 1, it increases gradually over time. As a result, the cumulative
multiplier we obtain (1.6) is substantially larger than 1.
6
See Table 1 in the online appendix that can be found as supplementary material at
http://dx.doi.org/10.1016/j.red.2013.01.004.
7
Although the DEFK model focuses mainly on resaleability constraints, borrowing constraints also play a signi…cant role in generating large multipliers in this model. If there
are no borrowing constraints, as discussed in Kiyotaki and Moore (2008), new investment could be wholly …nanced by issuing new equity. As a result, shocks to resaleability
constraints would have negligible impacts.
7
worth to note that to generate a liquidity trap, Carrillo and Poilly (2013)
assume that the capital returns perceived by entrepreneurs are a¤ected by
a risk premium shock, as in Smets and Wouters (2007). Since the empirical
relevance of this kind of shock is uncertain (see Chari, Kehoe and McGrattan (2009) for a detailed discussion), our approach o¤ers an alternative way
to generate a liquidity trap in a …nancial crisis. Despite the di¤erence in
our approach, our …ndings are consistent with those of the above-mentioned
papers, further strengthening their conclusions.
The remainder of this paper is structured as follows: Section 2 describes
the details of the DEFK model. Section 3 discusses the calibration of parameters. Section 4 compares the …scal multipliers produced by the DEFK
model to those obtained using a standard DSGE model in various scenarios.
In Section 4, we also include the results of the sensitivity analysis of several
parameters. Section 5 concludes our …ndings.
2
The Model with Credit Frictions
The model that we use in our analysis is proposed by DEFK (2011), which
incorporates a speci…c form of credit frictions as discussed in Kiyotaki and
Moore (2008). Households in this model are liquidity constrained and face
stochastic shocks that further tighten their liquidity. Government expenditure is absent in DEFK (2011). We introduce the role of exogenous government expenditure to the model for the study of the …scal multiplier. Other
aspects of the model are standard New Keynesian (see, for example, CEE
and SW). Details of the model are discussed below.
2.1
Households
The economy consists of a continuum of identical households. Each household
consists of a continuum of members j 2 [0; 1]. In each period, members have
an i.i.d. opportunity { to invest in capital. Household members (j 2 [0; {))
8
who receive the opportunity to invest are “entrepreneurs”, whereas those
who do not (j 2 [{; 1]) are “workers”. Entrepreneurs invest and do not work.
Workers work to earn labour income. Each household’s assets are divided
equally among its own members at the beginning of each period. After members …nd out whether they are entrepreneurs or workers, households cannot
reallocate their assets. If any household member needs extra funds, they
need to obtain them from external sources. This assumption is important
as it gives rise to liquidity constraints. At the end of each period, household
members return all their assets plus any income they earn during the period
to the asset pool.8
The representative household’s utility depends on the aggregate consumpR1
tion Ct
Ct (j) dj as consumption goods are jointly utilised by its mem0
bers. Each member seeks to maximise the utility of the household as a whole,
which is given by:
Et
1
X
s=t
s t
Cs1
1
1
1+
Z
1
Hs (j)1+ dj ,
(1)
{
where is the discount factor, is the coe¢ cient of relative risk aversion, and
is the inverse Frisch elasticity of labour supply. Labour supply Ht (j) = 0
for entrepreneurs. Each period, household members choose optimally among
non-durable consumption, saving in bonds or equity and, if they are entrepreneurs, investment in capital. Details of their saving and investment
options are as follows: (i) Investment in new capital. Entrepreneurs have
the opportunity to invest in new capital (It ) which costs pIt per unit. Each
unit of capital goods generates a rental income of rtk , depreciates at a rate
of and has a market value of qt . The return on new capital is therefore
8
The assumption that entrepreneurs and workers belong to the same household is based
on Shi (2011). This is di¤erent from the setting in KM (2008), in which entrepreneurs and
workers are two separate entities. As noted by DEFK (2012), adopting this assumption
increases the ‡exibility of the model to incorporate various modi…cations for sensitivity
analysis.
9
k +(1
rt+1
)qt+1
.
pIt
Entrepreneurs can borrow to invest. Borrowing is in the form
of issuing equity, NtI , that entitles the holder to claim the future returns on
the underlying capital goods. (ii) Saving in government bonds. Household
members can save in risk-free government bonds, Lt , which have a unit face
value and pay a gross nominal interest rate, Rt , over the period t to t + 1.
(iii) Saving in private equity. Household members can also purchase the equity issued by other households, NtO , at the market price of qt . As equity
holders receive income from the underlying capital goods, the return on eqrk +(1 )q
uity over t to t + 1 is t+1 qt t+1 .9 The household’s net equity is de…ned
as its equity holdings plus its capital stocks minus any equity issued by it:
Nt NtO + Kt NtI .
At the beginning of each period, the household also receive dividends
from intermediate-goods and capital-goods …rms amounting to Dt and DtK
respectively. The household pay lump-sum taxes, t , to the government.
Taxes are lump-sum so that they are non-distortive. The intertemporal budget constraint is:10
Ct +
pIt It
+ qt [Nt
It ] + Lt =
rtk
+ (1
) qt Nt
1
+
Rt 1 Lt
t
+Dt +
DtK
t
1
+
Z
1
{
Wt (j)
Ht (j) dj
Pt
(2)
where t PPt t 1 is the gross in‡ation rate at t and Wt (j) is the nominal wage
earned by type-j workers. Entrepreneurs and workers face di¤erent problems
as explained below.
9
The implicit assumption is that holding the equity issued by other households has the
same risk level as holding the capital goods directly.
10
In this paper, stock variables at t show the amounts of stocks at the end of the period.
This is di¤erent from the timing convention of stock variables in DEFK (2011). In their
paper, stock variables at t are de…ned as the amounts at the beginning of the period.
10
2.1.1
Entrepreneurs
In the steady state and the post-shock equilibria, the market price of equity
qt is always greater than the investment cost of new capital pIt . Hence, the
rk +(1 )q
return on new capital ( t+1 pI t+1 ) is strictly greater than the return on
rk +(1 )q
( t+1 qt t+1 )
t
which is the same as the real return on government
equity
bonds due to the anti-arbitrage condition. Entrepreneurs are rational, so
they would invest all their available resources in new capital. To spare more
funds for investment, entrepreneurs do not spend on consumption goods, i.e.,
Ct (j) = 0 for j 2 [0; {). They would also sell all their bond holdings so that
Lt (j) = 0 for j 2 [0; {). There are, however, constraints if entrepreneurs
want to obtain funds through equity: (i) Borrowing constraint. Entrepreneurs can borrow by issuing equity of only up to 2 (0; 1) fraction of their
new investment. (ii) Resaleability constraint. In each period, entrepreneurs
can sell only up to t 2 (0; 1) fraction of their net equity holdings. Since
borrowing and resaleability constraints are both binding, entrepreneurs’net
equity evolves according to Nt (j) = (1
) Nt 1 (j) + (1
)It (j).
t ) (1
Combining entrepreneurs’ …rst order conditions for Ct (j), Lt (j) and Nt (j)
with the intertemporal budget constraint (2) gives the aggregate investment:
It =
Z
{
It (j) dj = {
rtk + (1
) qt
t
Nt
1
pIt
0
+
Rt
1 Lt 1
t
+ Dt + DtK
t
qt
(3)
Investment expenditure depends on the abundance of the household’s liquidity. By contrast, in a standard DSGE model without credit frictions, investment opportunity is not scarce. Investment in new capital simply provide
the same rate of return as other forms of assets. In such models, investment
expenditure is una¤ected by liquidity conditions.
A credit crisis occurs when there is a sudden worsening of equity’s resaleability, expressed by a 60% drop in the resaleability parameter t from
11
its steady-state value, . Evolution of t thus follows bt = et < 0, where
t
bt
. Unlike DEFK who assume that bt follows a two-state Markov
process, we assume that bt stays below zero following a credit shock for a
deterministic number of periods.11
2.1.2
Workers
After solving for entrepreneurs’problem, the workers’consumption and saving decisions can be derived by considering the household as a whole. Workers
choose Ct , Lt and Nt to maximise the household’s utility (1) subject to the
intertemporal budget constraint (2) and the investment decision of entrepreneurs (3). The …rst-order conditions give the respective Euler equations for
bonds and equity:
Ct
Ct
=
=
Et
Et
(
Ct+1
(
Ct+1
"
"
Rt
t+1
pIt+1
qt+1
{ qt+1
+ I
pt+1
k
rt+1
+ (1
qt
) qt+1
Rt
t+1
#)
{ qt+1
+ I
pt+1
(4)
k
pIt+1 rt+1
+ (1
) qt+1
qt
qt+1
These Euler equations reduce to the standard ones when { = 0. In the
DEFK model, there is a premium on top of the standard returns on bonds
and equity because households are credit-constrained. By choosing to buy
one extra unit of government bonds at t instead of consumption, the bondt
extra units of liquidity at t + 1: Similarly, by choosing to
holder gains Rt+1
purchase one extra unit of equity at t instead of spending, the equity-holder
rk +(1 )qt+1 t+1
receives t+1
extra units of liquidity at t + 1. The extra liquidity
qt
allows them to pro…t from investment opportunity if it arrives at t + 1.
The wage- and price-setting assumptions in this model are standard New
11
Carlstrom, Fuerst and Paustian (2012) …nd that an interest rate peg with stochastic
exit tends to exaggerate the value of the …scal multiplier. In this model, although the
duration of a zero-bound interest rate is endogenous, it depends heavily on the duration of
the credit crisis that causes the zero-bound condition. Therefore, we assume that a credit
crisis lasts for a deterministic number of periods.
12
t+1
#)
(5)
Keynesian. Workers supply di¤erentiated labour to the production sector
through the arrangement of employment agencies, who bundle di¤erentiated
labour supply into homogeneous units for …rms to hire. Wages are negotiated
by labour unions representing each speci…c type of workers. Labour unions
enjoy some degree of monopoly power which allows them to set wages in
a staggered basis. Intermediate-goods …rms choose the optimal amounts of
labour and capital inputs that maximise their expected pro…ts, taking wages
and capital rent as given. They set prices for their di¤erentiated products
according to the Calvo-pricing scheme, and sell them to …nal-goods …rms for
the production of homogeneous …nal goods. Capital-goods producers convert
…nal goods into physical capital, incurring an adjustment cost. Details of
these standard features are included in the Appendix, together with some
equilibrium equations.
2.2
Government Policies
bt = G G
b t 1 + eG
Evolution of government spending follows G
t , where G governs the persistence of government spending. A government spending shock
bt Gt G , where G and Y are
is measured as a percentage of GDP so that G
Y
the respective steady-state values of government spending and output. The
government’s budget constraint is:
Gt +
R t 1 Lt
1
=
t
(6)
+ Lt
t
In addition, the …scal rule requires taxes to be proportional to the government’s debt at the beginning of each period:
t
=
Rt 1 Lt
1
RL
,
(7)
t
where
> 0. and RL are the respective steady-state values of taxes and
government debt. The value of
is low to re‡ect that the adjustment on
taxes is slow compared to bond issue, so the government has to obtain funds
13
for …scal expansion mainly by issuing bonds.
The central bank adopts a generalised Taylor rule similar to the one in
SW (2007), which targets both in‡ation and output. The nominal interest
rate also follows a short-run feedback from the change in output:
Rt = max
8
<
:
Rt R1 R
t
Yt
Y
Y
!1
R
Yt
Yt 1
Y
9
=
; 1
;
(8)
where R is the steady-state gross nominal interest rate, R is the interest
rate smoothing parameter,
> 1; and Y and
Y are both between
zero and one. The zero lower bound on the nominal interest rate requires
that Rt cannot be lower than 1. The gross real interest rate is obtained by
t
.
rt = Et (Rt+1
)
2.3
Equilibrium and Solution Strategy
The resource constraint of the economy requires that:
It
Yt = Ct + 1 + S( ) It + Gt
I
(9)
Other aggregate equilibrium equations are included in the Appendix. In
DEFK, equations are approximated around a steady state by log-linearisation.
Since in our simulation experiments the competitive equilibria achieved following a credit shock can stay far away from the steady state for a long
time, applying log-linearisation may lead to misleading results. For this reason, we instead carry out deterministic simulations with the exact, nonlinear
equations using Dynare. Under the deterministic setting, the model assumes
that agents have perfect foresight on the future paths of shocks and they
expect with certainty that no subsequent shock to follow in the future. The
deterministic simulations generated by Dynare are in fact the exact paths of
the endogenous variables that evolve according to the model’s equilibrium
14
equations and shock structure. The solutions obtained in this way are not
linearised, therefore provide better estimates than the log-linear solutions
when the economy is far away from the steady state.
3
Calibration
Most of the calibration in this paper is drawn from the estimations of SW, except for the parameters related to credit frictions which largely follow DEFK.
The calibrated values are summarised in Table 1. Two important parameters, the borrowing constraint and the resaleability constraint t , jointly
determine the amount of liquidity in the economy. We follow DEFK to set
the steady-state values of both and to 0.185, which means that entrepreneurs can sell up to 56% (= 1 0:8154 ) of their equity holdings in one
year’s time. Also following DEFK, a credit shock is modelled as a 60% drop
in the value of t from 0.185 to 0.074 (i.e., et = 0:6). In DEFK, is …xed
at its steady-state value even in a credit crisis. In our analysis, we also study
the e¤ects of a tightening of borrowing constraints by lowering the value of
from 0.185 to 0.074.
Other parameters related to capital investment are {, , and . Consistent with DEFK, we calibrate the i.i.d. opportunity to invest in each quarter
({) to 0.05, which equals to a 20% opportunity to invest in one year.12 The
capital adjustment cost parameter ( ) is set to 1 as in DEFK. The capital
share in the production function ( ) and the quarterly depreciation rate ( )
takes on the conventional values of 0.36 and 0.025 respectively.
For the parameters that are standard in a DSGE model, we assign values
mainly by referring to the mode of the posterior estimates obtained by SW.
12
As noted by DEFK, 5% is a conservative estimate of the investment opportunity in
the literature. We thus carried out numerical experiments to increase the value of { and
found that even a slight increase of { to 5.5% would cause the condition that qt > pIt not to
hold. Since such condition is crucial in deriving the …rst order conditions of entrepreneurs,
we stick with DEFK’s calibration to set { at 5%.
15
Structural parameters:
0.99 Discount factor
1.39 Relative risk aversion
0.025 Depreciation rate
0.36 Capital share
1
Capital goods adjustment cost parameter
1.92 Inverse Frisch elasticity of labour supply
0.11 Price mark-up
f
0.11 Wage mark-up
!
0.65 Price Calvo probability
p
0.73 Wage Calvo probability
!
Parameters related to liquidity constraints:
{
0.05 Probability of investment opportunity
0.185 Borrowing constraint at steady state
0.185 Equity resaleability constraint at steady state
Policy parameters:
2.03 Taylor rule coe¢ cient on in‡ation
0.08 Taylor rule coe¢ cient on output
Y
0.22
Taylor rule coe¢ cient on change in output
Y
0.81 Interest rate smoothing
R
0.80 Persistence of government spending
G
0.1
Taxation rule parameter
Table 1: Calibration
16
Consumption-to-GDP ratio
C=Y
0.60
Investment-to-GDP ratio
I=Y
0.22
Government spending share
G=Y
0.18
Quarterly GDP
Y
2.92
Quarterly labour
H
0.85
Capital stocks
K
25.84
Public debt-to-GDP ratio
L=4Y
0.40
Tax-to-GDP ratio
=Y
0.19
Real wage
w
1.97
k
Capital rent
r
3.66%
Cost of new capital
pI
1
Market price of equity
q
1.07
Real marginal cost
mc
0.90
Nominal interest rate (quarterly) R 1 0.57%
Real interest rate (quarterly)
r 1 0.57%
Table 2: Steady-state values of endogenous variables
The coe¢ cient of relative risk aversion ( ) is 1.39, and the inverse Frisch
elasticity of labour supply ( ) is 1.92. The Calvo probabilities for prices ( p )
and wages ( w ) are 0.65 and 0.73 respectively. Following Chari, Kehoe and
McGrattan (2000), we assume the curvature parameters of the Dixit-Stiglitz
aggregators in goods and labour markets to be 10, meaning a markup of 0.11
in both goods and labour markets. We set the discount factor ( ) equal to
0.99 as in DEFK.
We also adopt the estimates of SW to the values of the parameters governing the conduction of monetary policy. The coe¢ cients of in‡ation ( )
and output ( Y ) in the monetary policy rule are 2.03 and 0.08 respectively;
whereas the feedback coe¢ cient on the change in output ( Y ) is 0.22. The
degree of interest rate smoothing is calibrated at 0.81. As in DEFK, we
assume the taxation rule parameter ( ) to be 0:1, implying that the adjustment of taxes to the government’s debt position is gradual. As the baseline,
we follow Christiano, Eichenbaum and Rebelo (2011) to set the persistence
of government spending ( G ) at 0.8. In the next section, we also compare
17
the results with G to 0.97, which is the estimate obtained by SW.
The steady-state values of the endogenous variables are reported in Table
2. Two steady-state ratios are exogenous: the public debt-to-GDP ratio
(L=4Y ) and the government spending share in GDP (G=Y ). The former
shows the amount of government bonds issued as a share of annual GDP.
Following DEFK, we set it to 40%. The latter takes the average value of
government spending share observed in the post-war United States of 18%.
In‡ation is zero at the steady state so that = 1.
4
How Large Is the Government Spending
Multiplier?
In the literature, studies of the …scal multiplier usually focus on the impact
dYt
, where dYt
Yt Y and dGt
Gt G
multiplier which is de…ned as dG
t
are the respective di¤erences of output and government expenditure from
their steady state at period t. As noted by Woodford (2011), this way of
calculating the multiplier requires the output rise to follow the same shape
of time path as that of the government spending rise for the multiplier to be
meaningful. We recognise in our simulations that the e¤ects of …scal stimulus
on GDP are often delayed, so the time paths of the two can di¤er from
each other substantially. For this reason, we instead focus on the cumulative
1
P
Et
multiplier, de…ned as
Et
dYt
t=0
1
P
. Under this de…nition, the multiplier measures
dGt
t=0
the expected cumulative increase in output given a one-dollar cumulative
increase in government expenditure. If it is greater than one, it implies that
any change in government spending has a spillover e¤ect on GDP. We study
the …scal multipliers under two scenarios: at normal times and in times of a
credit crisis. We de…ne normal times as the times when government spending
shocks are the only source of disturbances; credit crisis times are when the
18
economy is also hit by credit shocks.
4.1
The Multiplier at Normal Times
We use the DEFK model to calculate the …scal multiplier at normal times
by giving the steady state a positive government spending shock of 1% of
GDP. Credit frictions are present in the DEFK model even at normal times,
due to the borrowing and resaleability constraints faced by households. We
assume that a government spending shock follows an AR(1) process with a
persistence of 0.8 and that no subsequent shock is expected. We obtain a
cumulative output multiplier of 1.61.
How does this result compare with that obtained using a standard New
Keynesian DSGE model? We carry out a control experiment by stripping all
liquidity-constraint features from the DEFK model.13 With the same government spending shock, the model without credit frictions (henceforth the
“standard model”) predicts that the cumulative multiplier on output will
take a value of 0.55. This result is consistent with the conclusions reached
by Woodford (2011), who observes that the government spending multiplier
is less than one in a simple New Keynesian DSGE model in which monetary
policy follows a standard Taylor rule. In Table 3, we summarise the cumulative …scal multipliers obtained using the two models under normal economic
conditions.
To understand these results, we report in Figure 1 the impulse-response
functions (IRFs) of some key macroeconomic variables to a government spending shock. The IRFs obtained using the DEFK model are shown by solid
blue lines; those obtained using the standard model are shown by dotted
13
In this standard DSGE model, investment opportunities are not scarce, so { = 1:
Investing in capital is not more pro…table than investing in other assets as qt = pIt for all
t. The investment function (3) hence reverts to a standard Euler equation. There are no
liquidity constraints, implying that = t = 1. We use the calibration shown in Table 1
with the exception of , which is adjusted slightly to 0.9943 to keep steady-state interest
rates in line with those in the DEFK model.
19
Cumulative multipliers
Output Consumption Investment
Standard model
0.55
-0.35
-0.11
DEFK model
1.61
0.27
0.39
Table 3: Government spending multipliers on output, conusmption and investment
in normal times: the DEFK model vs. the standard model
black lines. The results show that the increase in government expenditures
creates in‡ation pressures, causing the central bank to tighten its monetary
policy. Using the model without credit frictions, both private investment
and consumption are crowded out by the rising interest rate. In addition,
forward-looking households anticipate the future tax increase and react by
reducing their consumption. As a result, the increase in output in this model
is moderate and short-lived. However, the IRFs generated by the DEFK
model are very di¤erent for some variables, especially investment. Following
the government spending shock, private investment falls slightly but then
rises in a hump-shaped manner after two quarters. The positive e¤ect on investment peaks at around ten quarters after the shock and disappears thirty
quarters after the shock.14 Consumption shows a similar hump-shaped pattern, rising above the steady state from quarter 10 onwards. It returns to
its steady-state value after about 80 periods from the shock. Accordingly,
the increase in output in the DEFK model is larger and more persistent. As
consumption and investment decrease in both models when the shock hits
the economy at t = 1, the impact multipliers on output are not too di¤erent
(0.70 in the DEFK model vs. 0.58 in the standard model). However, the
cumulative multiplier on output obtained using the DEFK model (1.61) is
almost three times that obtained using the standard model (0.55).
14
When the shock …rst hits the economy, investment decreases slightly. This is a consequence of our assumption that the stock variables in period t show the amounts of the
stocks at the end of the period. As a consequence, investment in period t depends on
entrepreneurs’bond holdings in period t 1. Thus, an increase in bond holdings in period
t has an e¤ect on investment in period t + 1.
20
Our impulse response analysis suggests that government spending expansion has positive spillover e¤ects on consumption and investment in the
DEFK model. To con…rm this suggestion, we compute the cumulative multipliers on consumption and investment in both the standard and the DEFK
models. These multipliers measure the expected cumulative increases in consumption and investment respectively, given a one-dollar cumulative increase
in government spending. Indeed, as shown in Table 3, both the investment
and the consumption multipliers are positive in the DEFK model. The consumption multiplier is 0.27 and the investment multiplier is around 0.4. Both
of these multipliers are negative in the standard model so that the cumulative output multiplier is less than one. The prediction by the standard model
that consumption decreases in response to a government spending expansion
is inconsistent with the empirical evidence provided by Blanchard and Perotti (2002), Gali, Lopez-Salido and Vallés (2007) and others. These studies
suggest that an increase in government spending increases consumption. The
responses of consumption in the DEFK model better match this empirical
evidence. Although consumption in the DEFK model declines when the
shock hits the economy, it rises quickly and becomes positive after about 10
quarters of the shock. It remains above the steady state for a long time.
Even though the positive deviations from the steady state are small, they
are highly persistent. Therefore, as the above-mentioned empirical studies
suggest, the consumption multiplier is positive.
There are two mechanisms through which …scal policy a¤ects overall output in the DEFK model. The …rst mechanism is the conventional one in
which public demand matters by a¤ecting real aggregate demand. We de…ne this mechanism as the “demand e¤ect” of …scal policy. As government
spending increases, it creates real aggregate demand which drives up output.
The second mechanism is through the provision of liquidity due to an increase in the supply of government bonds, which we de…ne as the “liquidity
e¤ect” of …scal policy. In the DEFK model, households are liquidity con-
21
strained in a way that entrepreneurs want to obtain funds to make pro…table
investments but cannot. The government, on the other hand, is not bound
by liquidity constraints. As the government issue a bond to a household to
be repaid by higher taxes on the household in the future, the government is
in e¤ect borrowing on behalf of the household at the risk-free interest rate.
For this reason, a …scal expansion …nanced mainly by bonds generates extra liquidity to the households. Such a correlation between public debt and
private liquidity has been discussed in Woodford (1990). The improvement
in liquidity is re‡ected by the reduction in the spread between liquid and
illiquid assets, which is de…ned as:
Et
k
rt+1
+ (1
qt
) qt+1
Rt
t+1
Our model shows that the quarterly spread reduces by 3 basis points following
the government spending expansion.
To con…rm the liquidity e¤ect of …scal policy, we carry out an experiment
to isolate the liquidity e¤ect of the government spending shock by shutting
down the demand e¤ect. We assume in this experiment that any increase
in government spending does not increase aggregate demand, so that the
resources constraint becomes:
It
Yt = Ct + 1 + S( ) It + G;
I
where G is the steady-state government spending. Given the same government spending shock and the same amount of government bonds issued as in
the previous case, we obtain the cumulative multipliers on output, consumption and investment solely due to the liquidity e¤ect. The results are shown
in Table 4.
Both the consumption and the investment multipliers by the liquidity
e¤ect are positive, suggesting that private consumption and investment are
22
Cumulative multipliers due to liquidity e¤ect
Output
Consumption
Investment
0.89
0.54
0.41
Table 4: Government spending multiplers in the DEFK model in normal times due
to the liquidity e¤ect
crowded in by an improvement in liquidity. The intuition for the crowding
in e¤ects is as follows: Government spending expansion in the DEFK model
is …nanced mainly by public debt since tax adjustments are slow. As the
government increases their spending, higher real interest rates and future
tax burdens cause households to reduce their consumption and increase their
bond holdings. This improves households’liquidity since government bonds
are more liquid than private equity. As described in the model, investment
in new capital is more pro…table than holding government bonds or private
equity. Therefore, when investment opportunity arrives, utility-maximising
entrepreneurs sell all their bond holdings and obtain extra liquidity to invest
in new capital. Investment thus increases following the government spending
expansion.15
The increase in investment has a knock-on e¤ect on consumption. The
fact that consumption becomes positive later than investment reinforces this
insight (see Figure 1). The intuition for the positive multiplier on consumption due to the liquidity e¤ect is as follows. Due to intertemporal substitution e¤ects, rising interest rates cause workers to respond to the government
spending shock initially by reducing consumption and increasing labour supply. As we assume that government spending follows an AR(1) process with a
15
Following Shi (2011), DEFK assume that entrepreneurs and workers in a household
pool their assets at the beginning of each period. When pooling is not allowed, as in
Kiyotaki and Moore (2008), entrepreneurs and workers are separate entities and the opportunity for entrepreneurs to invest is scarce. In that version of the model, an increase in
government borrowing would increase the bond holdings of non-investing entrepreneurs.
This would provide investing entrepreneurs with more liquidity when investment opportunity arrives. Therefore, without the asset-pooling assumption, the DEFK model still
suggests a large multiplier e¤ect on investment.
23
persistence parameter of 0.8, the increase in government spending dissipates
over time. As government spending falls, the real interest rate decreases.
Workers hence increase their consumption and reduce the labour supply. As
capital is still being produced, re‡ected by the persistently higher than usual
level of investment, the demand for labour is greater than steady state. A
greater demand for labour translates into higher real wages. The resulting
increase in real wages leads to an increase in consumption. Indeed, as the
IRF …gures show, consumption closely follows the dynamics of real wages.
Combining the demand and the liquidity e¤ects gives us the overall e¤ects
of …scal policy that we show earlier in Table 3. The large …scal multiplier on
output in the DEFK model at the value of 1.61 suggests that …scal policy is
highly e¤ective in stimulating growth in a liquidity constrained economy.
Due to the presence of credit constraints, Ricardian equivalence does not
hold in the DEFK model. Changes in taxes do a¤ect households’behaviour
and the value of the …scal multiplier is expected to be sensitive to the taxation
rule. We carry out sensitivity analysis on the taxation rule parameter, ,
which measures how quickly the government increases taxes following bond
issues. In the baseline,
is set to 0.1 following DEFK to re‡ect that a slow
rise in taxes. If we increase
to 1, the cumulative multiplier on output in
the DEFK model reduces to 0.67. As one would expect, this model indicates
that the government should delay increasing taxes to ensure e¤ective …scal
policy.
The stickiness of prices and wages also plays a role in generating a large
…scal multiplier. Table 5 presents the cumulative multipliers on output we
obtain with di¤erent degrees of nominal rigidities given the same government
spending shock. Column i ( p;w = 0) shows the results under fully ‡exible
prices and wages. Absent both price and wage stickiness, the standard model
gives a very low cumulative output multiplier of 0.09. The DEFK model
suggests a much larger multiplier (0.90), although it is small compared to the
baseline case (1.61). These results seem to suggest that, in the DEFK model,
24
Cumulative multipliers on output
under di¤erent degrees of nominal rigidities
(i) p;w = 0 (ii) w = 0
(iii) p = 0
Standard model
0.09
0.16
0.51
DEFK model
0.90
0.97
1.59
Table 5: Government spending multipliers under (i) fully ‡exible prices and wages;
(ii) sticky prices and ‡exible wages; and (iii) ‡exible prices and sticky
wages
both credit frictions and nominal rigidities play a key role in generating large
…scal multipliers.
An impulse-response analysis (not reported here) reveals the reason why
the …scal multipliers are smaller without nominal rigidities. In the case with
fully ‡exible prices and wages, government spending expansion causes in‡ation to quickly increase by more than it does in the case with sticky prices
and wages. The nominal interest rate therefore increases by more, as the
central bank adopts a Taylor rule that targets in‡ation. The high degree of
interest rate smoothing implies that the nominal interest rate stays high for
a long time. Since in‡ation is not persistent under ‡exible prices and wages,
in‡ation expectations fall quickly after the shock, causing the real interest
rate to stay persistently high along with the nominal interest rate. As a
consequence, the crowding out e¤ects of …scal policy are ampli…ed, giving
smaller …scal multipliers in this case.
Table 5 also reports the results with only one kind of nominal rigidity.
Column ii ( w = 0) shows the results obtained with fully ‡exible wages
but sticky prices; Column iii ( p = 0) shows those obtained with sticky
wages and fully ‡exible prices. With price stickiness alone, the multipliers
are not too di¤erent from those obtained absent nominal rigidities ( p;w =
0). With wage stickiness alone, on the other hand, we are able to obtain
multipliers very similar to those in the baseline case, both in the DEFK
and the standard models. These results imply that wage stickiness is more
25
important than price stickiness in producing a large …scal multiplier, a …nding
which is consistent with previous studies that suggest wage stickiness causes
more output persistence than price stickiness.16
The …ndings reported in Christiano et al. (2011) and Woodford (2011)
suggest that the …scal multiplier is smaller as the parameter governing the
persistence of government spending ( G ) increases. We repeat our experiments to …nd out the multipliers in both models by increasing G from 0.8
to 0.97, which is the estimate suggested by SW. The cumulative output multiplier in the DEFK model reduces to 1.04 in this case, whereas the one in
the standard model falls to only 0.27. The reason for this result is that as
the government spending rise is more persistent, the present value of the
associated tax rises also increases, causing larger negative wealth impacts
on consumption. The rise in output is therefore much smaller, resulting in
much a smaller …scal multiplier. However, our conclusion that the multiplier
is larger in the DEFK model than in the standard model remains unchanged.
We also carry out sensitivity analysis on the monetary policy rule. In
this experiment, we assume that, instead of (8), the central bank follows a
standard Taylor rule with
= 1:5, Y = 0:125 and no interest rate inertia.
In this case, the cumulative multiplier on output in the DEFK model is
slightly higher at 1.8, whereas the one in the standard model (0.6) is almost
the same as the baseline. These numbers seem to con…rm that the …scal
multiplier is larger the DEFK model regardless of the monetary policy rule.
4.2
The Multiplier at Times of Crisis
We have shown that the government spending multiplier is large in the presence of credit frictions, even without a credit shock. We now examine the
value of this multiplier at times of credit crisis. As mentioned earlier, a credit
crisis occurs when the value of the resaleability constraint parameter, t ; falls
16
See, e.g., Andersen (1998), Huang and Liu (1998) and Woodford (2003, Chapter 3)
for a discussion.
26
by 60%. The credit crisis brings about a liquidity trap. If the government
decides to increase its spending during a credit crisis, we assume that this
happens in the same quarter as the arrival of the credit shock, i.e., t = 1.
The cumulative …scal multiplier on output in a credit crisis is obtained by
1
P
Et
(dYt dYt )
t=0
Et
1
P
, where dYt denotes the change in output from steady state due
dGt
t=0
to the combined e¤ects of the credit shock and the government spending
shock, and dYt denotes the same due to the credit shock alone by holding
Gt constant. The di¤erence between these two measures the output change
that is speci…cally due to the increase in government spending. The multipliers on consumption and investment are calculated in the same way, with
Yt being replaced by Ct and It respectively.
Using the DEFK model, we simulate credit crises of various expected durations, and compute the cumulative multipliers in response to a government
spending shock of 1% of GDP with G = 0:8.17 This exercise cannot be carried out using the standard model as it does not allow for …nancial frictions.
Table 6 shows our results. We also report in the table the number of periods
in which the nominal interest rate falls to zero. Our results suggest that the
longer is the credit crisis, the longer the liquidity trap is. In addition, the
longer is the liquidity trap, the larger the …scal multiplier is. The DEFK
model implies values of the cumulative output multiplier between 2.00 and
2.28 in the crisis state, which are much higher than the value in normal times.
To determine the cause of a larger multiplier in the crisis state, we report
the IRFs of a credit crisis with an expected duration of three years, both
17
The size of the government spending shock is the same as that in the …rst section of
Cogan et al. (2010). Erceg and Linde (2012) …nd that the value of the multiplier can
be a¤ected by the size of the …scal stimulus when the liquidity trap is endogenous. The
larger is the …scal stimulus, the faster the economy exits the liquidity trap, causing a
smaller multiplier. We repeat our experiments by increasing the size of the shock to 2%
of GDP. We …nd that at normal times, the multipliers are una¤ected; at times of crisis,
the multipliers decrease only slightly (by around 0.1 on average).
27
Duration of Duration of
Cumulative multipliers
credit crisis liquidity trap Output Consumption Investment
1q
1q
2.00
0.68
0.32
4q
3q
2.09
0.78
0.32
8q
6q
2.17
0.86
0.32
12q
10q
2.22
0.91
0.34
16q
14q
2.27
0.95
0.34
20q
18q
2.28
0.97
0.34
Table 6: Government spending multipliers on output, conusmption and investment
in times of crisis in the DEFK model: Baseline case
with and without …scal stimulus.18 Figure 2 displays the IRFs for investment, capital, government bonds and government spending; Figure 3 plots
the remainder of the key variables. We …rst discuss the e¤ects of the liquidity shock in the case without …scal expansion. The shock leads to a large
decrease in the resaleability of equity, which means that entrepreneurs can
obtain fewer funds for investment by selling their equity. As it is clear from
Equation (3), the reduction in equity resaleability leads to a large fall in investment. Figure 2 shows that the fall in investment in period t = 1, when
the shock …rst hits the economy, is as large as 19%. This substantial fall
in investment seems to suggest that in the DEFK model, most new capital
investment is …nanced by the sales of entrepreneurs’ asset holdings, rather
than the issues of new equity. Consumption, output and employment fall
by signi…cant amounts when the credit shock hits. In period 1, both output and consumption fall by around 10%, while labour hours fall by around
15%.19 Re‡ecting the ‡ight to liquidity, households’bond holdings increase
18
Note that the IRFs are not smooth in this case. Most of the lines bend upwards after
12 quarters from the shock, when the economy is expected to exit from the credit crisis.
19
The fall in economy activity we obtain here is more severe than that suggested by
DEFK (2011). In DEFK (2011), the government carries out quantitative easing in a
credit crisis by buying private assets and selling government bonds in the open market.
Such policy improves liquidity in the economy and help alleviate the adverse e¤ects of
a credit shock. In our paper, we focus our study on the e¤ectiveness of …scal policy.
Therefore, to simplify our model, we assume that no quantitative easing is carried out in
28
by around 4% in period 1 and continue to rise in a hump shape. As the
…gure shows, the nominal interest rate hits its zero lower bound in response
to the credit shock. The nominal interest rate is zero-bound for ten quarters.
In‡ation decreases by around 3.7%, and because of the zero-bound nominal
interest rate, the real interest rate increases by around 2%.
These responses describe the initial e¤ects of the credit shock. Figure 2
also shows that investment follows the path of the equity resaleability closely.
As the reduction in the equity resaleability is assumed to be persistent, the
e¤ect on investment is also persistent. Investment stays ‡at and far below the
steady state after the arrival of the credit shock until the equity resaleability returns to its normal value. Unlike investment, consumption responds
gradually to the credit shock as its dynamics are largely governed by the real
interest rate, which adjusts gradually. As the real interest rate declines over
time, consumption gradually increases.
We now consider the crisis case with …scal expansion. As at normal times,
the increase in public demand and the improvement in liquidity following
the government spending expansion lead to an increase in real aggregate
demand. As a result, in‡ation shows a smaller decrease. Given the zerobound nominal interest rate and the lower de‡ation, the real interest rate
increases by less relative to the case without …scal stimulus, leading to smaller
falls in consumption and hence in output. A natural question arises: why
is the …scal multiplier larger in the crisis state than in normal times? The
main reason is that the multiplier e¤ect on consumption is larger in a liquidity
trap. To con…rm this, we also report in Table 6 the cumulative multipliers on
consumption and investment in the DEFK model. Indeed, the consumption
multiplier is larger than that in normal times and increases substantially as
the liquidity trap lengthens, whereas the investment multiplier is similar to
that in normal times (see Table 3).
To gain an insight into the role that credit constraints play in generating
a crisis.
29
Duration of
Cumulative multipliers
liquidity trap Output Consumption Investment
1q
1.65
0.47
0.18
4q
1.84
0.61
0.23
8q
1.66
0.48
0.18
12q
1.42
0.30
0.12
16q
1.24
0.17
0.08
20q
1.13
0.09
0.05
Table 7: Government spending multipliers on output, conusmption and investment
in the standard model with an imposed zero bound
large …scal multipliers in a liquidity trap, we also calculate the cumulative
multipliers on consumption, investment and output in a zero-bound state
using the standard model. Since credit frictions are absent in the standard
model, we cannot use it to simulate a credit crisis in the same way as we do
with the DEFK model. Instead, we follow Cogan et al. (2010) to assume
that the nominal interest rate in the standard model remains constant at
its steady-state value for various durations. The multipliers obtained are
reported in Table 7.
A comparison of the results in Tables 6 and 7 suggests that the …scal
multiplier is still larger in the DEFK model than that in the standard model
when the nominal interest rate is bound at zero due to the larger multipliers
on both consumption and investment in the DEFK model. In the standard
model, the value of the …scal multiplier is driven mainly by the multiplier
on consumption. The investment multiplier is very small. We …nd that as
the crisis prolongs, the …scal multiplier in the standard model increases in
a hump-shaped manner, reaching its peak when the zero-bound state lasts
for one year. This …nding is related to the observation by Christiano et al.
(2011) and Woodford (2011), who suggest that the …scal multiplier is largest
if the …scal expansion lasts exactly as long as the zero-bound state. Since
we assume that government spending evolves according to an AR(1) process
30
Normal times:
DEFK model
Cumulative multipliers
Output Consumption Investment
1.76
0.36
0.40
Crisis times:
Duration of Duration of
Cumulative multipliers
credit crisis liquidity trap Output Consumption Investment
1q
1q
1.81
0.42
0.39
4q
2q
2.09
0.79
0.30
8q
5q
2.18
0.90
0.29
12q
9q
2.25
0.97
0.30
16q
13q
2.32
1.04
0.30
20q
17q
2.36
1.08
0.30
Table 8: Government spending multipliers on output, consumption and investment in the DEFK model with lower steady-state value of the borrowing
constraint parameter (theta = 0.074)
with a persistence parameter of 0.8, the majority of the public spending
rises in our experiments occurs within the …rst four quarters after the shock.
The …scal multiplier is largest when the zero-bound state lasts for a similar
duration. As the liquidity trap lengthens, …scal policy becomes less e¤ective
and the multiplier decreases.
4.3
Borrowing Constraints
So far in our analysis, we have kept the borrowing constraint parameter, ,
constant at its steady-state value even in times of credit crisis. Recall that
represents the maximum amount entrepreneurs can borrow in each period
to fund their new investments. In reality, the di¢ culty to borrow varies
across economies as well as across industries. In light of these variations,
we seek to determine whether a change in the steady-state value of a¤ects
the value of the …scal multiplier. In the previous cases, we follow DEFK
31
Duration of Duration of
Cumulative multipliers
credit crisis liquidity trap Output Consumption Investment
1q
1q
2.00
0.68
0.32
4q
4q
2.10
0.81
0.30
8q
7q
2.13
0.84
0.31
12q
11q
2.18
0.88
0.32
16q
15q
2.18
0.89
0.33
Table 9: Government spending multipliers on output, consumption and investment
in the DEFK model in crisis times where borrowing and resaleability
constraints tighten simultaneously
(2011) to set equals to 0:185. Here, we lower its value to 0:074 to simulate
an economic environment with tougher borrowing conditions. This value is
chosen to be equal to the resaleability constraint parameter t in a credit
crisis.20 We calculate the …scal multipliers in both normal and crisis times
using the DEFK model and report our results in Table 8. We …nd that when
is lowered, the multipliers are not too di¤erent from those in the baseline
case. Although the multipliers in normal times are slightly bigger (e.g. 1.76
as opposed to 1.61 for the output multiplier), those obtained for the crisis
cases are almost the same as those in the Table 6. This is probably because
of the fact that, as we mention above, the majority of new investment is
…nanced by selling asset holdings, rather than issuing new equity.
One may argue that the borrowing conditions should be worsening instead
of static in a credit crisis. In view of this argument, we redo the simulation
experiments by assuming that the value of t is dynamic and decreases along
with t in a credit crisis. A credit crisis is thus re-de…ned as when the
borrowing and the resaleability constraints tighten simultaneously, in a way
that t and t fall hand-in-hand from their steady-state values to their low
values of 0.074. With the same government spending shock, the cumulative
20
Changing the value of from 0.185 to 0.074 requires us to also change the capital
share in the production function, , from 0.36 to 0.275 in order to keep the steady-state
real interest rate the same as in the baseline case for fair comparison.
32
multipliers on output, consumption and investment are obtained and the
results are shown in Table 9.21 The values of the multipliers seem to be very
close to those in the baseline case (see Table 6). Therefore, our conclusion
from the previous experiment remains unchanged.
5
Summary and Conclusions
In this paper, we have extended the DEFK model by introducing a role for
government spending. We use the resulting model to study the e¤ects of
an increase in government expenditure on the macroeconomy. The DEFK
model accounts for liquidity constraints and generates a liquidity trap in a
credit crisis.
Our main …nding is that …scal policy can be highly e¤ective in an economic environment in which private …nancial assets are partially liquid and
government bonds are liquid. There are two channels causing this result.
First, an increase in public spending stimulates output by increasing real
aggregate demand. Second, a bond-…nanced …scal expansion increases the
proportion of liquid assets in private sector wealth through an increase in
the supply of government bonds. As investment opportunities in physical
capital are particularly attractive relative to holding …nancial assets, credit
constrained entrepreneurs sell all their asset holdings to obtain funds for new
capital investment when investment opportunity arrives. Hence, the improvement in liquidity caused by …scal expansion gives rise to a positive e¤ect on
private investment. The increase in investment increases consumption and
output, generating a large …scal multiplier.
We also study the e¤ects of …scal policy in a liquidity crisis. A negative shock to liquidity reduces the resaleability of private assets further and
21
In this experiment, we …nd that the market value of equity is not always higher than
the cost of capital when the credit crisis lasts for 5 years. Since the …rst order conditions
of the DEFK model are derived based on the condition that qt > pIt , we do not include
the results obtained for a 5-year crisis from this experiment.
33
brings about a liquidity trap. As the multiplier e¤ect on consumption is
larger when the nominal interest rate is bound at zero, the …scal multiplier
is even larger under a negative liquidity shock. This result is consistent with
previous research …ndings which suggest that, relative to the case without
…scal expansion, an increase in public demand at the zero lower bound pushes
up prices, thereby lowers the real interest rate and stimulates consumption
and output (see, for example, Christiano, Eichenbaum and Rebelo (2011)).
In sum, our …ndings suggest that government spending may be a powerful
tool to stimulate output in the short-run in an economic environment in which
private …nancial assets are only partially liquid.
34
A
Appendix
A.1
Standard New Keynesian Features of the Model
Di¤erentiated workers j 2 [{; 1] supply labour Ht (j) to the production sector
through the arrangement of employment agencies as in Erceg, Henderson
and Levin (2000). Competitive employment agencies choose their pro…tmaximising amount of Ht (j) to hire, taking nominal wages Wt (j) as given.
They combine Ht (j) into homogeneous units of labour input, Ht , according
to:22
#1+ !
"
!
Z 1
1+ !
1
1
Ht (j) 1+ ! dj
Ht =
1 {
{
Accordingly, the demand for type-j labour is:
Ht (j) =
1
1
Wt (j)
{
Wt
1+ !
!
Ht ,
where !
0 and Wt is the aggregate wage index. Each type-j labour is
represented by a labour union who sets their nominal wage Wt (j) optimally
on a staggered basis. Each period, there is a history-independent probability
of (1
! ) for a union to reset their wage. Otherwise, they keep their nominal
wage constant. The optimal wage-setting equation, which is the same across
labour unions, in real terms is:
Et
1
P
(
s=t
s t
!)
Cs
8
>
>
>
<w
et
>
t;s
>
>
:
1
1 {
(1 +
!)
w
et
t;s ws
Cs
1+ !
!
v
Hs
9
>
>
>
=
>
>
>
;
w
et
t;s ws
1+ !
!
Hs = 0,
(10)
The term 1 1{ is added to the labour aggregate to simplify the notations without
changing the substance.
22
35
where w
et (j)
ft (j)
W
Pt(
is the optimal wage chosen by a labour union at t,
1, for s = t
. The zero-pro…t condition
t+1
t+1 t+2 ::: s , for s
for employment agencies gives rise to the dynamics of wt :
wt
Wt
Pt
and
t;s
wt
1
!
= (1
et
!) w
1
!
+
1
!
wt
!
1
(11)
t
Firms are classi…ed according to the goods they produce. Final-goods
…rms produce homogeneous …nal goods Yt by combining heterogeneous interi1+ f
hR
1
1
, where f
0.
mediate goods Yt (i) according to Yt = 0 Yt (i) 1+ f di
Their pro…t-maximising condition implies that the demand for type-i inter1+ f
i f
h
Yt , where Pt (i) and Pt are the respecmediate good is Yt (i) = PPt (i)
t
tive nominal prices for intermediate and …nal goods. Monopolistic competitive intermediate-goods …rms produce according to the production function
Yt (i) = At Kt (i) Ht (i)1 , where At is productivity and is the capital share.
Intermediate-goods …rms maximise their real pro…ts Dt (i) by choosing the
optimal capital and labour inputs, taking real wage and rental rate of capt (i)
= (1 ) wrkt .
ital as given. The cost-minimising conditions imply that K
Ht (i)
t
Accordingly, their marginal cost is:
wt
1
mct = mct (i) =
At
1
1
rtk
,
(12)
which is universal across …rms. Intermediate-goods …rms also set nominal
prices for their heterogeneous goods. In each period, each …rm has a constant
probability of 1
p to reset their price. They keep their price unchanged
otherwise. Firms who reset their price choose the one that maximises their
expected future pro…ts, giving the following price-setting equation (in real
terms):
36
Et
1
P
s t
p
s=t
Cs
pet
(1 +
1+ f
f
pet
f ) mcs
t;s
Ys = 0,
(13)
t;s
e
Pt (i)
where pet (i)
as the optimal price chosen at t. Given the zero-pro…t
Pt
condition for …nal-goods …rms, we obtain the evolution of in‡ation:
1= 1
p
pet
1
f
+
1
p
1
f
(14)
t
Capital-goods …rms convert …nal goods into capital goods. The adjustment cost is quadratic in aggregate investment in a way that S( IIt ) =
2
It
1 , where I is the steady-state investment and is the adjustment
2 I
cost parameter. Under this equation, S(1) = S 0 (1) = 0 and S 00 (1) > 0.
Capital-goods …rms choose the amount of It to produce which maximises
their pro…ts DtK = pIt
1 + S( IIt ) It . The …rst-order condition is:
It It
It
pIt = 1 + S( ) + S 0 ( )
I
I I
A.2
(15)
Aggregate Equilibrium Conditions
Capital evolves according to:
Kt = (1
) Kt
1
+ It
(16)
R1
The market clears for both labour and capital such that Ht = 0 Ht (i)di and
R1
Kt 1 = 0 Kt (i)di. The aggregate capital-labour ratio becomes:
Kt 1
=
Ht
(1
and the aggregate production function is:
37
wt
,
) rtk
(17)
A t Kt
1
Ht 1
=
Z
1
Yt (i)di = Yt
0
Z
0
1
Pt (i)
Pt
1+ f
f
di.
(18)
Capital is owned by households through their private equity holdings:
(19)
Kt = Nt
The pro…ts for intermediate-goods and capital-goods …rms are wholly distributed to households as dividends. Substituting for Dt and DtK , (3) becomes:
It = {
rtk + (1
) qt
t
Nt
1
+ rt 1 L t
38
1
+ Yt wt Ht
pIt
qt
rtk Kt
1
+ pIt It
1 + S( IIt ) It
(20)
t
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43
Output
Consumption
Inv estment
0. 5
0. 4
0. 2
0
0
% from ss
% from ss
% from ss
0. 6
-0.2
-0.4
0
-0.5
10
20
30
40
20
Labour
40
60
80
20
Real wage
40
60
Capital
0. 1
0. 2
0. 5
0
10
20
30
40
% from ss
% from ss
% from ss
1
0
-0.1
-0.2
Inf lation
10
20
30
0. 1
0
-0.1
40
Nominal interest rate
20
40
60
80
Real interest rate
0. 1
0
20
30
40
0. 7
0. 6
0. 5
Tax
% from ss
% from ss
1
0
20
20
30
30
40
2
1
0
10
DEFK
20
30
0. 8
0. 7
0. 6
0. 5
40
Gov ernment bonds
2
10
10
10
20
30
40
Gov ernment spending
% of GDP from ss
10
0. 8
quarterly rate
quarterly rate
quarterly rate
0. 2
40
1
0. 5
0
10
20
30
40
standard
Figure 1: IRFs to a government spending shock in normal times: the DEFK model
vs. the standard model
44
Investment
Government bonds
0
7
6
5
% from ss
% from ss
-5
-10
4
3
2
-15
1
0
-20
5
10
15
20
25
5
Capital
10
15
20
25
20
25
Government spending
0
1
% of GDP from ss
% from ss
-1
-2
-3
-4
-5
0.8
0.6
0.4
0.2
0
5
10
15
20
-0.2
25
no fiscal stimulus
5
10
15
fiscal stimulus
Figure 2: IRFs to a three-year credit crisis: E¤ects of …scal stimulus
45
Output
Consumption
6
% from ss
-4
-6
-8
% from ss
-2
-2
% from ss
Tax
0
0
-4
-6
-8
-10
-10
5
10
15
20
25
Real wage
10
15
20
25
5
Inflation
0
-1
10
15
20
-1
-2
-3
-4
25
20
25
0.5
0
5
Real interest rate
10
15
20
25
5
10
15
20
25
15
20
25
φ
Labour
0
0
1
% from ss
2
% from ss
quarterly rate
15
Nominal interest rate
quarterly rate
quarterly rate
% from ss
1
10
1
0
2
5
2
0
5
3
-2
4
-5
-10
0
-20
-40
-60
5
10
15
20
25
-15
5
10
no fiscal stimulus
15
20
25
5
10
fiscal stimulus
Figure 3: IRFs to a three-year credit crisis: E¤ects of …scal stimulus
46